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Combinatorics and geometry of finite and infinite squaregraphs
"... Squaregraphs were originally defined as finite plane graphs in which all inner faces are quadrilaterals (i.e., 4cycles) and all inner vertices (i.e., the vertices not incident with the outer face) have degrees larger than three. The planar dual of a finite squaregraph is determined by a triangle ..."
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Cited by 12 (8 self)
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Squaregraphs were originally defined as finite plane graphs in which all inner faces are quadrilaterals (i.e., 4cycles) and all inner vertices (i.e., the vertices not incident with the outer face) have degrees larger than three. The planar dual of a finite squaregraph is determined by a trianglefree chord diagram of the unit disk, which could alternatively be viewed as a trianglefree line arrangement in the hyperbolic plane. This representation carries over to infinite plane graphs with finite vertex degrees in which the balls are finite squaregraphs. Algebraically, finite squaregraphs are median graphs for which the duals are finite circular split systems. Hence squaregraphs are at the crosspoint of two dualities, an algebraic and a geometric one, and thus lend themselves to several combinatorial interpretations and structural characterizations. With these and the 5colorability theorem for circle graphs at hand, we prove that every squaregraph can be isometrically embedded into the Cartesian product of five trees. This embedding result can also be extended to the infinite case without reference to an embedding in the plane and without any cardinality restriction when formulated for median graphs free of cubes and further finite obstructions. Further, we exhibit a class of squaregraphs that can be embedded into the product of three trees and we characterize those squaregraphs that are embeddable into the product of just two trees. Finally, finite squaregraphs enjoy a number of algorithmic features that do not extend to arbitrary median graphs. For instance, we show that mediangenerating sets of finite
Recognizing Partial Cubes in Quadratic Time
, 2007
"... We show how to test whether a graph with n vertices and m edges is a partial cube, and if so how to find a distancepreserving embedding of the graph into a hypercube, in the nearoptimal time bound O(n²), improving previous O(nm)time solutions. ..."
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Cited by 9 (2 self)
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We show how to test whether a graph with n vertices and m edges is a partial cube, and if so how to find a distancepreserving embedding of the graph into a hypercube, in the nearoptimal time bound O(n²), improving previous O(nm)time solutions.
On embeddings of CAT(0) cube complexes into products of trees via colouring their hyperplanes
 J. Combin. Theory Ser. B
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Shortest path problem in rectangular complexes of global nonpositive curvature
, 2012
"... CAT(0) metric spaces constitute a farreaching common generalization of Euclidean and hyperbolic spaces and simple polygons: any two points x and y of a CAT(0) metric space are connected by a unique shortest path γ(x, y). In this paper, we present an efficient algorithm for answering twopoint di ..."
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Cited by 3 (2 self)
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CAT(0) metric spaces constitute a farreaching common generalization of Euclidean and hyperbolic spaces and simple polygons: any two points x and y of a CAT(0) metric space are connected by a unique shortest path γ(x, y). In this paper, we present an efficient algorithm for answering twopoint distance queries in CAT(0) rectangular complexes and two of theirs subclasses, ramified rectilinear polygons (CAT(0) rectangular complexes in which the links of all vertices are bipartite graphs) and squaregraphs (CAT(0) rectangular complexes arising from plane quadrangulations in which all inner vertices have degrees ≥ 4). Namely, we show that for a CAT(0) rectangular complex K with n vertices, one can construct a data structure D of size O(n2) so that, given any two points x, y ∈ K, the shortest path γ(x, y) between x and y can be computed in O(d(p, q)) time, where p and q are vertices of two faces of K containing the points x and y, respectively, such that γ(x, y) ⊂ K(I(p, q)) and d(p, q) is the distance between p and q in the underlying graph of K. If K is a ramified rectilinear polygon, then one can construct a data structure D of optimal size O(n) and answer twopoint shortest path queries in O(d(p, q) log ∆) time, where ∆ is the maximal degree of a vertex of G(K). Finally, if K is a squaregraph, then one can construct a data structure D of size O(n logn) and answer twopoint shortest path queries in O(d(p, q)) time.